Zonal polynomials for wreath products
Hiroshi Mizukawa
National Defense Academy in Japan Department of Mathematics Yokosuka 239-8686 Japan Yokosuka 239-8686 Japan
DOI: 10.1007/s10801-006-0031-6
Abstract
The pair of groups, symmetric group S 2 n and hyperoctohedral group H n , form a Gelfand pair. The characteristic map is a mapping from the graded algebra generated by the zonal spherical functions of ( S 2 n , H n ) into the ring of symmetric functions. The images of the zonal spherical functions under this map are called the zonal polynomials. A wreath product generalization of the Gelfand pair ( S 2 n , H n ) is discussed in this paper. Then a multi-partition versions of the theory is constructed. The multi-partition version of zonal polynomials are products of zonal polynomials and Schur functions and are obtained from a characteristic map from the graded Hecke algebra into a multipartition version of the ring of symmetric functions.
Pages: 189–215
Keywords: keywords zonal polynomial; Schur function; Gelfand pair; Hecke algebra; hypergeometric function
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References
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3. N. Bergeron and A. Garsia, “Zonal polynomials and domino tableaux,” Discrete Math. 99 (1992), 3-15.
4. C. Curtis and I. Reiner, Methods of Representations Theory, vol. 1 A Wiley-Interscience, 1981.
5. I.M. Issacs, “Character theory of finite groups,” Dover Publications Inc, 1994.
6. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications 16 (1981).
7. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd. ed., Oxford, Clarendon Press, 1995.
8. H. Mizukawa, Zonal spherical functions on the complex reflection groups and ( m + 1, n + 1)- hypergeometric functions, Advances in Math. 184 (2004), 1-17.
9. H. Mizukawa and H Tanaka, “( n + 1, m + 1)-hypergeometric functions associated to character algebras,” Proc AMS 132(9) (2004), 2613-2618.
10. D. Rockmore, “Fast fourier transforms for wreath products,” Appl. Comput. Harmon. Anal. 2 (1995), 279-292.
11. R. Stanley “Some combinatorial properties of Jack symmetric functions,” Advances in Math. 77 (1989), 76-115.
12. J. Stembridge, “On Schur's Q-functions and the primitive idempotents of a commutative Hecke algebra,” J. Alg. Comb. 1 (1992), 71-95.
13. A. Takemura, Zonal Polynomials, Inst. of Math. Statistics Lecture Notes,-Monograph Series vol.
4. Hayward, California, 1984.
14. A. Zelevinsky, Representations of finite classical groups. A Hopf Algebra Approach, Lecture Notes in Mathematics,
869. Springer-Verlag, Berlin-New York, 1981.